In the first picture the Node is not deletable because it is part of a 2 Nodes shape only, deleting it will result with a 1 Node shape which causes problems so we prevent it. Like with a 2 Nodes segment, you can only delete the entire Path/object with the SELECT tool.

In the second picture, you have to first click on the path with the Edit-path tool to make it active (The Nodes and direction arrows will show up) only then edit the Nodes. I know it’s a bit of a hassle, but that’s a “coding necessity”.